The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of
integers
József Balogh, Hong Liu, Maryam Sharifzadeh,
Andrew Treglown*
University of Birmingham
20th September 2014
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Introduction
Definition
Denote [n] := {1, . . . , n}. A set S ⊆ [n] is sum-free if x + y 6∈ S for
every x, y ∈ S (x and y are not necessarily distinct).
Examples
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{4, 5, 8} is not sum-free.
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Set of odds is sum-free.
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{n/2+1, n/2+2,. . . , n} is sum-free.
Last two examples show there are at least 2n/2 sum-free subsets of
[n].
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Introduction
Cameron-Erdős Conjecture (1990)
The number of sum-free subsets of [n] is O(2n/2 ).
Green (2004), Sapozhenko (2003)
There are constants ce and co , s.t. the number of sum-free subsets
of [n] is
(1 + o(1))ce 2n/2 , or (1 + o(1))co 2n/2
depending on the parity of n.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Introduction
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The previous result doesn’t tell us anything about the
distribution of the sum-free sets in [n].
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In particular, recall that 2n/2 sum-free subsets of [n] lie in a
single maximal sum-free subset of [n].
Cameron-Erdős Conjecture (1999)
There is an absolute constant c > 0, s.t. the number of maximal
sum-free subsets of [n] is O(2n/2−cn ).
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Lower bound construction
There are at least 2bn/4c maximal sum-free subsets of [n].
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Suppose n is even. Let S consist of n together with precisely
one number from each pair {x, n − x} for odd x < n/2.
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Notice distinct S lie in distinct maximal sum-free subsets of
[n].
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Roughly 2n/4 choices for S.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Main result
Denote by f (n) the number of maximal sum-free subsets in [n].
Recall that f (n) ≥ 2bn/4c .
Cameron-Erdős Conjecture (1999)
∃c > 0,
f (n) = O(2n/2−cn ).
Luczak-Schoen (2001)
−28 n
f (n) ≤ 2n/2−2
for large n
Wolfovitz (2009)
f (n) ≤ 23n/8+o(n) .
Balogh-Liu-Sharifzadeh-T. (2014+)
f (n) = 2n/4+o(n) .
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Tools
From additive number theory:
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Container lemma of Green.
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Removal lemma of Green.
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Structure of sum-free sets by Deshouillers, Freiman, Sós and
Temkin.
From extremal graph theory: upper bound on the number of
maximal independent sets for
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all graphs by Moon and Moser.
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triangle-free graphs by Hujter and Tuza.
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Not too sparse and almost regular graphs.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Sketch of the proof
Container Lemma [Green]
There exists F ⊆ 2[n] , s.t.
(i) |F| = 2o(n) ;
(ii) ∀S ⊆ [n] sum-free, ∃F ∈ F, s.t. S ⊆ F ;
(iii) ∀F ∈ F, |F | ≤ (1/2 + o(1))n and the number of Schur
triples in F is o(n2 ).
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Sketch of the proof
Container Lemma [Green]
There exists F ⊆ 2[n] , s.t.
(i) |F| = 2o(n) ;
(ii) ∀S ⊆ [n] sum-free, ∃F ∈ F, s.t. S ⊆ F ;
(iii) ∀F ∈ F, |F | ≤ (1/2 + o(1))n and the number of Schur
triples in F is o(n2 ).
By (i) and (ii), it suffices to show that for every container A ∈ F,
f (A) ≤ 2n/4+o(n) .
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Constructing maximal sum-free sets
Removal+Structural lemmas ⇒ classify containers A ∈ F:
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Case 1: small container, |A| ≤ 0.45n;
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Case 2: ‘interval’ container, ‘most’ of A in [n/2 + 1, n].
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Case 3: ‘odd’ container, |A \ O| = o(n).
Moreover, in all cases A = B ∪ C where B is sum-free and
|C | = o(n).
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Constructing maximal sum-free sets
Removal+Structural lemmas ⇒ classify containers A ∈ F:
I
Case 1: small container, |A| ≤ 0.45n;
I
Case 2: ‘interval’ container, ‘most’ of A in [n/2 + 1, n].
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Case 3: ‘odd’ container, |A \ O| = o(n).
Moreover, in all cases A = B ∪ C where B is sum-free and
|C | = o(n).
Crucial observation
Every maximal sum-free subset in A can be built in two steps:
(1) Choose a sum-free set S in C ;
(2) Extend S in B to a maximal one.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
maximal sum-free sets ⇒ maximal independent sets
Definition
Given S, B ⊆ [n], the link graph of S on B is LS [B], where V = B
and x ∼ y iff ∃z ∈ S s.t. {x, y , z} is a Schur triple.
L2 [1, 3, 4, 5]
1
3
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
4
5
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
maximal sum-free sets ⇒ maximal independent sets
Definition
Given S, B ⊆ [n], the link graph of S on B is LS [B], where V = B
and x ∼ y iff ∃z ∈ S s.t. {x, y , z} is a Schur triple.
Lemma
Given S, B ⊆ [n] sum-free and I ⊆ B, if S ∪ I is a maximal sum-free
subset of [n], then I is a maximal independent set in LS [B].
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Case 1: small container, |A| ≤ 0.45n.
Recall A = B ∪ C , B sum-free, |C | = o(n).
Crucial observation
Every maximal sum-free subset in A can be built in two steps:
(1) Choose a sum-free set S in C ;
(2) Extend S in B to a maximal one.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Case 1: small container, |A| ≤ 0.45n.
Recall A = B ∪ C , B sum-free, |C | = o(n).
Crucial observation
Every maximal sum-free subset in A can be built in two steps:
(1) Choose a sum-free set S in C ;
(2) Extend S in B to a maximal one.
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Fix a sum-free S ⊆ C (at most 2|C | = 2o(n) choices).
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Consider link graph LS [B].
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Moon-Moser: ∀ graphs G , MIS(G ) ≤ 3|G |/3 .
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So # extensions in (2) is exactly MIS(LS [B]),
MIS(LS [B]) ≤ 3|B|/3 ≤ 30.45n/3 20.249n .
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In total, A contains at most 2o(n) × 20.249n 2n/4 maximal
sum-free sets.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Cases 2 and 3.
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Now container A could be bigger than 0.45n.
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This means crude Moon-Moser bound doesn’t give accurate
bound on f (A).
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Instead we obtain more structural information about the link
graphs.
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Cases 2 and 3.
I
Now container A could be bigger than 0.45n.
I
This means crude Moon-Moser bound doesn’t give accurate
bound on f (A).
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Instead we obtain more structural information about the link
graphs.
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For example, when A ‘close’ to interval [n/2 + 1, n] link
graphs are triangle-free
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Hujta-Tuza: MIS(G ) ≤ 2|G |/2 for all triangle-free graphs G .
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Gives better bound on f (A).
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers
The number of maximal sum-free subsets of integers
Thank you!
J. Balogh, H. Liu, M. Sharifzadeh, A. Treglown
The number of maximal sum-free subsets of integers